document.write( "Question 1112642: Divide: 1-4i/5-6i. Write your answer in a+bi form
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\n" ); document.write( "I'm really confused on the a+bi thing, as well as solving the fraction. I tried to cancel out the i's, but I think that's the opposite of what I'm supposed to do. \n" ); document.write( "
Algebra.Com's Answer #727700 by ikleyn(52754)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "  \"%281-4i%29%2F%285-6i%29\" = \r\n" );
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document.write( "It is standard problem on complex numbers. I will show you the standard way of solving it.\r\n" );
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document.write( "Multiply the numerator and the denominator by the complex number conjugate to that of denominator.\r\n" );
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document.write( "In your case the denominator is  5-6i  and  the conjugate number to it  is  5+6i.\r\n" );
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document.write( "So, we will multiply the numerator and denominator by (5+6i).\r\n" );
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document.write( "    Since we multiply the numerator and denominator by the same number, the value\r\n" );
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document.write( "    of our fraction remains the same.  So, we can continue from the above \r\n" );
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document.write( "= \"%281-4i%29%2F%285-6i%29\" . \"%285%2B6i%29%2F%285%2B6i%29\" = \"%28%281-4i%29%2A%285%2B6i%29%29%2F%28%285-6i%29%2A%285%2B6i%29%29\"\r\n" );
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document.write( "Now it is better (until you gain the necessary practice) to work separately with the numerator and denominator.\r\n" );
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document.write( "Numerator = (1-4i)*(5+6i) = 5 - 20i + 6i - 24*(i^2) = 5 + 24 - 14i = 29-14i    (((<<<---=== you remember, of course, that  i^2 = -1)\r\n" );
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document.write( "Denominator = (5-6i)*(5+6i) = 25 - 30i + 30i - 36*(i^2) = 25+36 = 61.    \r\n" );
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document.write( "    The modified denominator is the product of the complex number and its conjugate, so it is a REAL NUMBER.  \r\n" );
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document.write( "    It is why we multiplied by the conjugate number:  to get a real number in the denominator !\r\n" );
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document.write( "So, our fraction is  \"NUM%2FDEN\" = \"%2829-14i%29%2F61\" = \"29%2F61\" - \"%2814%2F61%29%2Ai\".\r\n" );
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document.write( "It is your final presentation of the given fraction as a complex number in the form  z = a + bi.\r\n" );
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document.write( "In your case  a = \"29%2F61\",  b = \"-14%2F61\".\r\n" );
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\n" ); document.write( "\n" ); document.write( "What I showed to you is the standard method solving such problems.\r
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\n" ); document.write( "\n" ); document.write( "Memorize it as a mantra.\r
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\n" ); document.write( "\n" ); document.write( "There is a bunch of introductory lessons on complex numbers\r
\n" ); document.write( "\n" ); document.write( "    - Complex numbers and arithmetical operations on them\r
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\n" ); document.write( "\n" ); document.write( "    - Addition and subtraction of complex numbers in complex plane\r
\n" ); document.write( "\n" ); document.write( "    - Multiplication and division of complex numbers in complex plane\r
\n" ); document.write( "\n" ); document.write( "    - Raising a complex number to an integer power\r
\n" ); document.write( "\n" ); document.write( "    - How to take a root of a complex number\r
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\n" ); document.write( "\n" ); document.write( "    - Solved problems on taking roots of complex numbers\r
\n" ); document.write( "\n" ); document.write( "    - Solved problems on arithmetic operations on complex numbers\r
\n" ); document.write( "\n" ); document.write( "in this site.\r
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\n" ); document.write( "\n" ); document.write( "Also,  you have this free of charge online textbook in ALGEBRA-II in this site\r
\n" ); document.write( "\n" ); document.write( "    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.\r
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\n" ); document.write( "\n" ); document.write( "The referred lessons are the part of this online textbook under the topic  \"Complex numbers\".\r
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\n" ); document.write( "\n" ); document.write( "Save the link to this textbook together with its description\r
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\n" ); document.write( "\n" ); document.write( "Free of charge online textbook in ALGEBRA-II
\n" ); document.write( "https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson\r
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\n" ); document.write( "\n" ); document.write( "into your archive and use when it is needed.\r
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